Analyze phase planes

Consider a first order ordinary differential equation,

Which of the following condition must be true such that one equilibrium point is a sink and other equilibrium point is a source?

In biotech laboratory, a new trait of bacterium is cultivated in a controlled environment. The growth of population is logistic with rate ‘0.5’ and carrying limit of ‘20’ units. If initially 3 units were added for the cultivation, which of phase line describe the population growth?

Which of the following autonomous ordinary differential equation has node on the equilibirium point meaining point is neither a source nor a sink?

Consider a differential equation whose equilibrium point is a node. A node is point where the system is semi-stable. 

Which of the modified equation has a equilibrium point for the stable system?

A charged capaitor ‘C’ is connected across a resistor ‘R’ in a circuit. The voltage across the capacitor is given by Vc(t). What is state of system at equilibrium?

In a chemical reaction, the concentration of reactant is decreasing exponentiallly as reaction proceeds. The phase line  for same is given as,

[X] represents the concentration of reaction. What is the maximum amount of reactant concetration that can be added for stable reaction and complete conversion of reactant to product ?

A population of shrimp in lake has exceeded the carrying capacity of lake. If the population growth is represented by the given phase line,

It is observed that growth of population is logistic with very high growth rate. Which of the following solution curve best represent the population of shrimp over time?

A system has two equilibrium points ‘a’ and ‘b’ with ‘a’ as source and ‘b’ as sink (b>a). A factor is changed in differential equation, which translate the system having only one equilibrium point as node ‘N’. Which of the following bifuraction diagram represent the translation?

Self cleaning property of lake is given as,

‘k’ is the amount of effluent added to the lake (k>1). What is the maximum value ‘k’ possible for the lake to clean itself and remain stable?

The possible solution curves of a mechanical system is given below with , and as the three equilibirum points.

Which of the following phase line correctly represent mechanical system?

Consider a predator prey system of fox and rabbits given as,

Rabbit :

Fox :

The above system has equilibrium point at origin. The solution of the system will reach this equilibrium, 

Consider a system that models the sale of mobile phones of two competitive brand A and B. The following represent the nullcline for the system.

Which is the equilibrium point where both brands survive in the market?

Consider a system of differential equation given as,

dx/dt=3x+9y

dy/dt=-ax-4y

If the phase plane with trajectories is given as,

What is the value of a?

An electrical system with oscillation is modelled as,

What is the state of system if the equilibrium lies at origin?

A local N95 mask manufacturer is having a surge of demand of mask due to pandemic outbreak. If the mask consumer demand and mask supply in market is modelled as,

If currently the supply of mask is lying below 5000, what will happen?